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Operator algebraic approach to topological phases

Periodic Reporting for period 1 - OATP (Operator algebraic approach to topological phases)

Reporting period: 2015-09-01 to 2017-08-31

It was long thought that all phases of matter could be described by what is called Landau’s theory of symmetry breaking. With the discovery of topological phases in the early 1980’s, this belief turned out to be false, and that there is a whole new class of phases of matter. What distinguishes these models is that fundamental physical characteristics are related to topological invariants of the system. Topology is the mathematical study of which shapes or geometric objects can be continuously deformed into each other, without “tearing” or breaking them. A topological invariant is then something that does not change under such a continuous deformation. An example is the number of holes in a surface: zero for a sphere and one for a donut. This number of holes does not change if one deforms the surface without tearing it.

This robustness under deformations has interesting physical consequences, since it implies that corresponding physical properties are also robust. As a consequence, these physical properties depend only qualitatively on the microscopic details of the system. This is particularly important because in practice, theoretical models are idealisations of real materials, and it is impossible to implement an experiment exactly.

Some of the most interesting applications of topological states of matter are in quantum computing. A quantum computer would provide huge benefits over ordinary “classical” computers in various computational tasks. Perhaps the best-known example is factorisation of numbers into prime numbers, but equally important is the simulation of, for example, complex molecules. This is particularly important in the search for new medicines.

A major obstacle in building a quantum computer is that they typically are very sensitive to noise from the environment. This noise can render the outcome useless. Although there are protocols to correct for errors introduced in this way, it is tremendously difficult to make the error rate low enough. Because of this, people have started looking for fault-tolerant quantum computers, which are inherently stable against noise from the environment. An important contender here are the systems with topological order: the idea is to use their topological properties for quantum computation. Since these topological properties are by their very nature stable against small perturbations of the system, this in principle can be used to provide fault tolerance.

Despite considerable progress on such systems, and the many examples that have been found, there still is a lot that is not clear. Some properties are believed to be true based on physical grounds, but the underlying mathematical principles are not clear. An example is the stability of such systems. Even though for specific systems there are good physics-based arguments on why they should be stable, a sound mathematical understanding is lacking. It is important to understand such fundamental properties better in a rigorous way, to be able to talk about many systems at the same time. In other words, to generalise these results to a wide class of systems.

In our project, we want to contribute to a better understanding of the mathematical underpinnings of such systems. We focus in particular on what are called quantum spin systems. Such systems are defined on a lattice (or “grid”), with on each point on the grid a simple quantum mechanical system. One can for example think of a solid, where atoms are localised in a regular pattern. The interesting thing about the topologically ordered systems that we are interested in, is that they support what are called “anyons”. These are excitations of the system that behave like particles with very special properties (which again have a topological origin). These anyons play a fundamental role in the proposals for fault-tolerant computing. In our project, we (i) provide a theory to derive the properties of these anyons, and (ii) study their stability. That is, if we
During the first two years we have addressed two topics in particular: stability of the properties of the anyons, and applications of the quantum dimension (a property of anyons) to quantum information theory. For the first topic, we managed to classify all ground states of abelian quantum double models, a particular model with topological order. Ground states are the lowest energy states, and play a fundamental role in the theory. These ground states are related to the different types of anyons the theory has, and a good understanding of the ground states provides an essential first step in studying the stability of them. Work on this is nearing completion, and will show that the anyons are indeed stable in this model.

The second topic is the application of the quantum dimension. We discovered that this quantity can be interpreted in terms of the amount of information one can hide from a malicious third party, using the anyons in a system. Also from a mathematical point of view this leads to very interesting structures, which open up new possibilities on the application of powerful advanced mathematical tools in quantum information theory.

The project so far has resulted into four publications, and an additional two nearing completion. One of the publications is a book, which provides graduate students and other researchers the necessary mathematical background of our work.
Our work so far has provided a better understanding of the mathematics of topologically ordered systems. In particular the stability result is one the first of its kind. In the remainder of the project we will address other aspects of the theory, in particular on systems which have a boundary. Clearly this is relevant in practice, since in an experiment all systems will be of finite size and hence have a boundary. It is also important physically, and we attempt to get a better understanding of how we can describe what happens at the boundary. This is the starting point to discuss related questions. For example, one can ask the same question about stability: how is the boundary affected if we perturb the system?

Another exciting development is the use of subfactor theory in quantum information. Subfactor theory is well studied in mathematics, and has led to surprising connections between seemingly distinct topics in physics and mathematics. However, applications to quantum information theory have not been considered thus far. More generally, most results in quantum information are for finite systems, while subfactors naturally appear in the description of systems with infinitely many degrees of freedom. The extension of quantum information theory to infinite systems is important because we know that there are many systems in nature that are best described as infinite systems.
Overview of the different topics and how they relate to each other.